Modulated Unit-Norm Tight Frames for Compressed Sensing

TL;DR

This paper introduces a new compressed sensing framework using modulated unit-norm tight frames that satisfy the RIP with fewer measurements, enabling more efficient and practical sensing models for applications like OFDM channel estimation.

Contribution

The paper proposes a unified structured sensing framework based on unit-norm tight frames, providing tighter measurement bounds and new models for practical compressed sensing applications.

Findings

RIP satisfied with O(s log^2 s log^2 n) measurements

Unified framework encompasses existing models and new structured sensing schemes

Application to OFDM channel estimation simplifies transceiver design

Abstract

In this paper, we propose a compressed sensing (CS) framework that consists of three parts: a unit-norm tight frame (UTF), a random diagonal matrix and a column-wise orthonormal matrix. We prove that this structure satisfies the restricted isometry property (RIP) with high probability if the number of measurements $m = O(s \log^2s \log^2n)$ for $s$-sparse signals of length $n$ and if the column-wise orthonormal matrix is bounded. Some existing structured sensing models can be studied under this framework, which then gives tighter bounds on the required number of measurements to satisfy the RIP. More importantly, we propose several structured sensing models by appealing to this unified framework, such as a general sensing model with arbitrary/determinisic subsamplers, a fast and efficient block compressed sensing scheme, and structured sensing matrices with deterministic phase modulations, all of which can lead to improvements on practical applications. In particular, one of the constructions is applied to simplify the transceiver design of CS-based channel estimation for orthogonal frequency division multiplexing (OFDM) systems.